The Boltzmann paradox is a reference to a thought experiment named after a scientist who was writing on thermodynamics and cosmology a century ago. Basically, it predicts that if the universe is made up of random structures coming together and falling apart, at any given point in time there should be more spontaneously created brains floating in deep space than there are intelligent observers like humans looking up at space from planets.
If even very large structures like galaxies ultimately emerge basically at random points in the universe out of gas and dust, it must be true that very small structures also emerge basically at random. In fact, the number of very small structures (little gas clouds, say) must be a lot higher than the number of very big structures (galaxies, say), for the same reason that you wouldn’t be surprised at rolling a six but you might be surprised at rolling six sixes in a row.
But the implication of this is that every once in a while, the random little object that spontaneously comes together won’t be a little gas cloud: it’ll be something with more random structure. Even something that looks exactly like a body, say. Or a brain. In the thought experiment, we call this a “Boltzmann brain”: a brain that has come together through random fluctuations of matter in the universe, out in deep space, rather than as part of a person on a planet.
This might seem incredibly unlikely, but in a big enough universe where structures are randomly coming together and falling apart all the time, even the incredibly unlikely must happen once in a while.
And there comes the paradox part. In a sufficiently huge universe, there must be a very large number of Boltzmann brains at any given moment in time. In fact, at any given point in time, there must be more “Boltzmann brains” (randomly created brains floating in space) than there are brains in the heads of actual intelligent observers of the universe, like ourselves.